Problem 6

Determine all functions

f:\mathbb{R}\to\mathbb{R}

such that

f(x-f(y))=f(f(y))+xf(y)+f(x)-1

for all real numbers

x,y

Step 2 of 5: Obtain all real differences

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In plain words

The nonzero slope makes image differences surjective.

c\ne0,\quad f(x-c)-f(x)=xc+f(c)-1

Detailed analysis

Setting $y=0$ gives $f(x-c)-f(x)=xc+f(c)-1$ . We must have $c\ne0$ , since $c=0$ would make the original equation read $f(x)=f(x)-1$ . As $x$ varies, the right side takes every real value, so every real number is a difference of two image values: $f(\mathbb R)-f(\mathbb R)=\mathbb R$ .